CDF

The cumulative distribution function (CDF) of a random variable X is a function F defined by F(x) = P(X ≤ x) for all real x. It completely characterizes the distribution of X; for a vector X = (X1, ..., Xk) the CDF is F(x1, ..., xk) = P(X1 ≤ x1, ..., Xk ≤ xk). The CDF is nondecreasing and right-continuous.

Key properties: lim x→-∞ F(x)=0, lim x→∞ F(x)=1. If X is discrete, F has jumps at points in the support corresponding to the probability masses; if X has a density f, then F is differentiable almost everywhere and F' = f, so F(x) = ∫_{-∞}^x f(t) dt.

Quantiles and inverse: If F is continuous and strictly increasing, its inverse F^{-1}(p) gives the p-th quantile. For mixed distributions, F is continuous except at points with mass. The CDF determines the distribution uniquely.

Examples: For a standard normal distribution, F is the standard normal CDF; for a Bernoulli(p) variable, F(x) = 0 for x < 0, F(x) = 1-p for 0 ≤ x < 1, and F(x) = 1 for x ≥ 1; for an exponential(λ) distribution, F(x) = 1 − e^{-λx} for x ≥ 0.

Empirical CDF and applications: The empirical CDF F_n(x) = (1/n) ∑ I{X_i ≤ x} is a nonparametric estimate of F. It converges to F almost surely as n grows. CDFs are used in hypothesis testing, goodness-of-fit, and in sampling methods such as inverse transform sampling.